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Modules

A homalg based package for the Abelian category of finitely presented modules over computable rings

2024.12-01

22 December 2024

Thomas Bächler
Email: thomas@momo.math.rwth-aachen.de
Homepage: http://wwwb.math.rwth-aachen.de/~thomas/
Address:
Thomas Bächler
Lehrstuhl B für Mathematik
RWTH Aachen
Templergraben 64
52062 Aachen
Germany

Mohamed Barakat
Email: mohamed.barakat@uni-siegen.de
Homepage: https://mohamed-barakat.github.io
Address:
Walter-Flex-Str. 3
57072 Siegen
Germany

Florian Diebold
Email: diebold@mathematik.uni-kl.de
Address:
Department of Mathematics
University of Kaiserslautern
67653 Kaiserslautern
Germany

Sebastian Gutsche
Email: gutsche@mathematik.uni-siegen.de
Homepage: https://sebasguts.github.io
Address:
Department Mathematik
Universität Siegen
Walter-Flex-Straße 3
57072 Siegen
Germany

Markus Lange-Hegermann
Email: markus.lange-hegermann@hs-owl.de
Homepage: https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/
Address:
Markus Lange-Hegermann
Hochschule Ostwestfalen-Lippe
Liebigstraße 87
32657 Lemgo
Germany

Vinay Wagh
Email: waghoba@gmail.com
Homepage: http://www.iitg.ernet.in/vinay.wagh/
Address:
E-102, Department of Mathematics,
Indian Institute of Technology Guwahati,
Guwahati, Assam, India.
PIN: 781 039.
India

Copyright

© 2007-2015 by Mohamed Barakat and Markus Lange-Hegermann This package may be distributed under the terms and conditions of the GNU Public License Version 2 or (at your option) any later version.

Acknowledgements

We are very much indebted to Alban Quadrat.

Contents

1.1 What is the role of the Modules package in the homalg project?

  1.1-1 Modules provides ...
  1.1-2 Rings supported in a sufficient way
  1.1-3 Principal limitation
  1.1-4 Ring dictionaries (technical)
  1.1-5 The advantages of the outsourcing concept
  1.1-6 Does this mean that homalg has only algorithms for the generic case?
  1.1-7 The principle of least communication (technical)
  1.1-8 Frequently asked questions

1.2 This manual

2 Installation of the Modules Package

3.1 Why are all examples in this manual over ℤ or ℤ/mℤ?

3.2 gap> ExamplesForHomalg();

3.3 A typical example

4.1 Ring Maps: Attributes

4.1-1 KernelSubobject
4.1-2 KernelEmb

4.2 Ring Maps: Operations and Functions

5.1 Relations: Categories and Representations

  5.1-1 IsHomalgRelations
  5.1-2 IsHomalgRelationsOfLeftModule
  5.1-3 IsHomalgRelationsOfRightModule
  5.1-4 IsRelationsOfFinitelyPresentedModuleRep

5.2 Relations: Constructors

5.3 Relations: Properties

5.3-1 CanBeUsedToDecideZeroEffectively
5.3-2 IsInjectivePresentation

5.4 Relations: Attributes

5.5 Relations: Operations and Functions

6.1 Generators: Categories and Representations

  6.1-1 IsHomalgGenerators
  6.1-2 IsHomalgGeneratorsOfLeftModule
  6.1-3 IsHomalgGeneratorsOfRightModule
  6.1-4 IsGeneratorsOfModuleRep
  6.1-5 IsGeneratorsOfFinitelyGeneratedModuleRep

6.2 Generators: Constructors

6.3 Generators: Properties

6.3-1 IsReduced

6.4 Generators: Attributes

6.4-1 ProcedureToReadjustGenerators

6.5 Generators: Operations and Functions

7.1 Modules: Category and Representations

  7.1-1 IsHomalgModule
  7.1-2 IsFinitelyPresentedModuleOrSubmoduleRep
  7.1-3 IsFinitelyPresentedModuleRep
  7.1-4 IsFinitelyPresentedSubmoduleRep

7.2 Modules: Constructors

  7.2-1 LeftPresentation
  7.2-2 RightPresentation
  7.2-3 HomalgFreeLeftModule
  7.2-4 HomalgFreeRightModule
  7.2-5 HomalgZeroLeftModule
  7.2-6 HomalgZeroRightModule
  7.2-7 \*
  7.2-8 Subobject
  7.2-9 Subobject
  7.2-10 LeftSubmodule
  7.2-11 RightSubmodule

7.3 Modules: Properties

  7.3-1 IsCyclic
  7.3-2 IsHolonomic
  7.3-3 IsReduced
  7.3-4 IsPrimeIdeal
  7.3-5 IsPrimeModule

7.5 Modules: Operations and Functions

  7.5-1 HomalgRing
  7.5-2 ByASmallerPresentation
  7.5-3 \*
  7.5-4 SubobjectQuotient

8.1 Maps: Categories and Representations

  8.1-1 IsHomalgMap
  8.1-2 IsHomalgSelfMap
  8.1-3 IsMapOfFinitelyGeneratedModulesRep

8.2 Maps: Constructors

  8.2-1 HomalgMap
  8.2-2 HomalgZeroMap
  8.2-3 HomalgIdentityMap

8.3 Maps: Properties

8.4 Maps: Attributes

8.5 Maps: Operations and Functions

8.5-1 HomalgRing
8.5-2 PreInverse

9 Module Elements

9.1 Module Elements: Category and Representations

9.1-1 IsHomalgElement
9.1-2 IsElementOfAModuleGivenByAMorphismRep

9.2 Module Elements: Constructors

9.3 Module Elements: Properties

9.3-1 IsElementOfIntegers

9.4 Module Elements: Attributes

9.5 Module Elements: Operations and Functions

9.5-1 HomalgRing

11 Symmetric Algebra and Koszul Complex

11.1 Symmetric Algebra: Constructor

11.1-1 SymmetricPower

11.2 Symmetric Algebra: Properties and Attributes

  11.2-1 IsSymmetricPower
  11.2-2 SymmetricPowerExponent
  11.2-3 SymmetricPowerBaseModule

12 Exterior Algebra and Koszul Complex

12.1 Exterior Algebra: Constructor

12.1-1 ExteriorPower

12.2 Exterior Algebra: Properties and Attributes

  12.2-1 IsExteriorPower
  12.2-2 ExteriorPowerExponent
  12.2-3 ExteriorPowerBaseModule

12.3 Exterior Algebra: Element Properties

12.3-1 IsExteriorPowerElement

12.4 Exterior Algebra: Element Operations

  12.4-1 Wedge
  12.4-2 ExteriorPowerElementDual
  12.4-3 SingleValueOfExteriorPowerElement

12.5 Koszul complex and Cayley determinant

  12.5-1 KoszulCocomplex
  12.5-2 CayleyDeterminant
  12.5-3 Gcd_UsingCayleyDeterminant

13.3 TorExt-Grothendieck

A The Mathematical Idea behind Modules

B Logic Subpackages

B.1 LIMOD: Logical Implications for Modules

B.2 LIHOM: Logical Implications for Homomorphisms of Modules

C Overview of the Modules Package Source Code

C.1 Relations and Generators

C.2 The Basic Objects

C.3 The High Level Homological Algorithms

C.4 Logical Implications for homalg Objects

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